{"title":"具有最大维纳指数的图形方向","authors":"Zhenzhen Li, Baoyindureng Wu","doi":"10.1002/jgt.23090","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we study the Wiener index of the orientation of trees and theta-graphs. An orientation of a tree is called no-zig-zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> achieving the maximum Wiener index is no-zig-zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski, and Tepeh conjectured that among all orientations of the theta-graph <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>Θ</mi>\n \n <mrow>\n <mi>a</mi>\n \n <mo>,</mo>\n \n <mi>b</mi>\n \n <mo>,</mo>\n \n <mi>c</mi>\n </mrow>\n </msub>\n </mrow>\n <annotation> ${{\\rm{\\Theta }}}_{a,b,c}$</annotation>\n </semantics></math> with <span></span><math>\n <semantics>\n <mrow>\n <mi>a</mi>\n \n <mo>≥</mo>\n \n <mi>b</mi>\n \n <mo>≥</mo>\n \n <mi>c</mi>\n </mrow>\n <annotation> $a\\ge b\\ge c$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>b</mi>\n \n <mo>></mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $b\\gt 1$</annotation>\n </semantics></math>, the maximum Wiener index is achieved by the one in which the union of the paths between <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>u</mi>\n \n <mn>1</mn>\n </msub>\n </mrow>\n <annotation> ${u}_{1}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>u</mi>\n \n <mn>2</mn>\n </msub>\n </mrow>\n <annotation> ${u}_{2}$</annotation>\n </semantics></math> forms a directed cycle of length <span></span><math>\n <semantics>\n <mrow>\n <mi>a</mi>\n \n <mo>+</mo>\n \n <mi>b</mi>\n \n <mo>+</mo>\n \n <mn>2</mn>\n </mrow>\n <annotation> $a+b+2$</annotation>\n </semantics></math>, where <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>u</mi>\n \n <mn>1</mn>\n </msub>\n </mrow>\n <annotation> ${u}_{1}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>u</mi>\n \n <mn>2</mn>\n </msub>\n </mrow>\n <annotation> ${u}_{2}$</annotation>\n </semantics></math> are the vertex of degree 3. We confirm the validity of the conjecture.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 3","pages":"556-580"},"PeriodicalIF":0.9000,"publicationDate":"2024-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Orientations of graphs with maximum Wiener index\",\"authors\":\"Zhenzhen Li, Baoyindureng Wu\",\"doi\":\"10.1002/jgt.23090\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we study the Wiener index of the orientation of trees and theta-graphs. An orientation of a tree is called no-zig-zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math> achieving the maximum Wiener index is no-zig-zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski, and Tepeh conjectured that among all orientations of the theta-graph <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>Θ</mi>\\n \\n <mrow>\\n <mi>a</mi>\\n \\n <mo>,</mo>\\n \\n <mi>b</mi>\\n \\n <mo>,</mo>\\n \\n <mi>c</mi>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation> ${{\\\\rm{\\\\Theta }}}_{a,b,c}$</annotation>\\n </semantics></math> with <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>a</mi>\\n \\n <mo>≥</mo>\\n \\n <mi>b</mi>\\n \\n <mo>≥</mo>\\n \\n <mi>c</mi>\\n </mrow>\\n <annotation> $a\\\\ge b\\\\ge c$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>b</mi>\\n \\n <mo>></mo>\\n \\n <mn>1</mn>\\n </mrow>\\n <annotation> $b\\\\gt 1$</annotation>\\n </semantics></math>, the maximum Wiener index is achieved by the one in which the union of the paths between <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>u</mi>\\n \\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation> ${u}_{1}$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>u</mi>\\n \\n <mn>2</mn>\\n </msub>\\n </mrow>\\n <annotation> ${u}_{2}$</annotation>\\n </semantics></math> forms a directed cycle of length <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>a</mi>\\n \\n <mo>+</mo>\\n \\n <mi>b</mi>\\n \\n <mo>+</mo>\\n \\n <mn>2</mn>\\n </mrow>\\n <annotation> $a+b+2$</annotation>\\n </semantics></math>, where <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>u</mi>\\n \\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation> ${u}_{1}$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>u</mi>\\n \\n <mn>2</mn>\\n </msub>\\n </mrow>\\n <annotation> ${u}_{2}$</annotation>\\n </semantics></math> are the vertex of degree 3. We confirm the validity of the conjecture.</p>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"106 3\",\"pages\":\"556-580\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-03-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23090\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23090","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this paper, we study the Wiener index of the orientation of trees and theta-graphs. An orientation of a tree is called no-zig-zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree achieving the maximum Wiener index is no-zig-zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski, and Tepeh conjectured that among all orientations of the theta-graph with and , the maximum Wiener index is achieved by the one in which the union of the paths between and forms a directed cycle of length , where and are the vertex of degree 3. We confirm the validity of the conjecture.
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The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
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